On the chain rule for the divergence of BV-like vector fields: applications, partial results, open problems

Ambrosio, L; De Lellis, C; Malý, J (2007). On the chain rule for the divergence of BV-like vector fields: applications, partial results, open problems. In: Berestycki, H; Bertsch, M; Browder, F E; Nirenberg, L; Peletier, L; Véron, L. Perspectives in nonlinear partial differential equations. Providence, RI: American Mathematical Society, 31-67.

Abstract

We discuss the problem of computing the distributional divergence of a vector field of the form h(w)B, where h is a smooth scalar function and B is a BV vector field, knowing the distributional divergence of all vector fields w_iB, where w_i are the components of w.

We present partial results on this problem, conjectures, and links with other problems related to the SBV regularity of solutions of Hamilton-Jacobi equations and systems of conservation laws, and a conjecture recently made by Bressan

Abstract

We discuss the problem of computing the distributional divergence of a vector field of the form h(w)B, where h is a smooth scalar function and B is a BV vector field, knowing the distributional divergence of all vector fields w_iB, where w_i are the components of w.

We present partial results on this problem, conjectures, and links with other problems related to the SBV regularity of solutions of Hamilton-Jacobi equations and systems of conservation laws, and a conjecture recently made by Bressan